zeb haradon wrote:
>
> I missed the beginning of this thread, but it seems like you're talking
> about whether the Goldbach conjecture could fall into the class of
> undecidable theorems. I don't know if anyone has pointed this out yet, but
> the Goldbach conjecture cannot fall into that class.
The beginning of the thread,
http://www.lucifer.com/exi-lists/extropians/1811.html
described an article in the March 10 issue of _New Scientist_ that
contains an informal description of the work of mathematician Gregory
Chaitin of IBM's Thomas J. Watson Research Center, who is interested in
randomness and its implications for the rest of mathematics, especially
the provability of assertions in number theory. The article mentions a
number Chaitin defined called Omega which is apparently an embodiment of
randomness, and other numbers called Super Omegas which exemplify some sort
of hierarchy of randomness.
Later posts by Samantha Atkins, Eliezer, and others harrumphed at
the article, e.g.:
http://www.lucifer.com/exi-lists/extropians/1811.html
The only mention I see in the _New Scientist_ article of anything that
even looks like the Goldbach Conjecture (not by name) or the Riemann
Hypothesis (by name) is the paragraph:
"Take the problem of perfect odd numbers. A perfect number has divisors
whose sum makes the number. For example, 6 is perfect because its
divisors are 1, 2 and 3, and their sum is 6. There are plenty of even
perfect numbers, but no one has ever found an odd number that is
perfect. And yet, no one has been able to prove that an odd number can't
be perfect. Unproved hypotheses like this and the Riemann hypothesis,
which has become the unsure foundation of many other theorems (New
Scientist, 11 November 2000, p 32) are examples of things that should be
accepted as unprovable but nonetheless true, Chaitin suggests. In other words,
there are some things that scientists will always have to take on trust."
Jim F.
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